The dynamics of sea level in tidal basins are one of the most addressed topics in physical oceanography. Theoretical prediction of tidal motion was pioneered by the application of a Fourier analysis by Lord Kelvin (Thomson, 1880) and later improved by Doodson (1921), who developed the tide-generating potential in harmonic form. Analysis and interpretation of tidal observations by Proudman and Doodson (1924) enhanced the understanding of sea-level fluctuations due to winds and changes in atmospheric pressure. The development of numerical 2D storm surge models by Peeck et al. (1983) and Flather and Proctor (1983) led to early warning systems for coastal flooding. With increasing computational power and the availability of satellite data, sea-level predictions have been continuously improved. However, current model predictions are not always perfect (Stanev et al., 2015a; Sandery and Sakov, 2017; Staneva et al., 2016; Ponte et al., 2019; De Mey-Frémaux et al., 2019; Jacobs et al., 2021), which emphasizes the need for further understanding of sea level.

A recent important evolution in predicting sea level in the North Sea was achieved in the framework of the development of the northwest European shelf forecasting system (e.g., O'Dea et al., 2012; Tonani et al., 2019) by enhancing the model resolution to 1.5 km. Thus, dynamical features such as coastal currents, fronts, and mesoscale eddies are better resolved, and the model results are improved, especially when compared to high spatial–temporal resolution observations.

Satellite altimetry has added critical information in the last 30 years (Madsen et al., 2015). Notably, different measurement techniques have different advantages and disadvantages. Satellite-derived sea-level information, which has revolutionized oceanography and climate science, particularly in addressing global and large-scale change, is of limited use when addressing near-coastal short-periodic variability. However, advancements are underway, and new satellite missions characterized by better spatial and temporal sampling have paved the way for improvements in coastal sea-level research (e.g., Dieng et al., 2021; Prandi et al., 2021; Dodet et al., 2020; Sanchez-Arcilla et al., 2021).

Tide gauge stations operating along the North Sea coast provide high-quality records of sea-level observations over a long period (Wahl et al., 2013). Ponte et al. (2019), reviewing the state of the science of coastal sea-level monitoring and prediction, outlined the importance of sea-level observations for studying sea-level variability. However, tidal gauges do not provide information about the basin-wide patterns of sea level. Furthermore, some of these data are not continuous; different gauges do not always operate simultaneously, and there are gaps in many of the records. Therefore, a combination of numerical model results with tide gauge measurements would be beneficial. A similar exercise was undertaken recently by Madsen et al. (2019) for the Baltic Sea and by Zhang et al. (2020) for the North Sea. Zhang et al. (2020) used machine learning to reconstruct the sea-level variability in the North Sea using observations from 19 coastal tide gauges and data from numerical models. Notably, they concluded that a relatively short-time record contains the most representative characteristics of sea-level dynamics in the North Sea. While this was clear for tides, it was not so obvious for changes in sea level caused by the atmosphere.

The North Sea (Fig. 1a) is a shallow sea with an average depth of ∼90 m located at the European continental shelf. The dynamics of sea level in the North Sea, which is the region of our study, can be considered a coupled response to different forcings, such as barotropic tides, wind, and atmospheric pressure, as well as forcings from the open boundaries and rivers, including a thermohaline forcing. The coupling of the respective processes is, in most cases, nonlinear (Jacob and Stanev, 2017), therefore, one cannot easily identify the response to individual drivers in isolation. This happens when either oscillatory motions have large amplitudes, e.g., tidal currents approaching 1 m s-1, or when wind-driven current is of the same order. Thus, there is a need to use methods tailored to detect and reproduce nonlinear dynamics. Nonlinear processes are difficult to predict, even with sophisticated models; therefore, one needs to identify situations in which predictions fail (Ponte et al., 2019). Furthermore, it is interesting to understand the reasons for the deviation in the model predictions from the observations. The detection of such situations in data from a network of tide gauges is the aim of the present study.

The spatial and temporal correlations of tide gauge data mirror the complex sea-level dynamics in the North Sea (Fig. 2). First, tidal wave propagation is influenced by the topography of the North Sea. The north–south decrease in water depth shifts the central amphidromic point southwards (see Fig. 1a for the oscillation pattern of the semidiurnal M2 tide). Two additional amphidromic systems are generated as a result of the superposition of incident and reflected Kelvin waves and cross-shore Poincaré waves with centers in the north, off the Norwegian coast, and in the south, between Suffolk and Holland (Fig. 1a). The amplitude of the tidal wave increases towards the British coast. Off the Danish and Norwegian coasts, the amplitude of the wave is significantly smaller due to dissipation by bottom friction in the shallow southern part of the North Sea (Fig. 2a).

The circulation of the North Sea and thus the sea level is forced also by the atmosphere through either wind stresses (Fig. 2c) or atmospheric pressure gradients (Fig. 2d). Considering the length scales and depth of the North Sea, the wind has a dominant effect. The prevailing westerly winds result in a dominant cyclonic circulation. A reversal of circulation caused by easterly winds seldom occurs (Stanev et al., 2019). For northwesterly and southeasterly winds, circulation may stagnate. As shown by Jacob and Stanev (2017), the interaction between tidal and wind-driven currents is nonlinear in areas of strong currents, particularly in the German Bight, and the atmospheric variability affects the spring–neap variability more strongly than the M2 tide. Additionally, shallow-water tides show significant small-scale spatial variability patterns in this region (Stanev et al., 2015b). Thus, in the southern North Sea, tidal forcing strongly impacts the residual circulation (Fig. 2b). In summary, the specific topography of the North Sea together with nonlinear effects lead to complex circulation patterns that hamper sea-level prediction.

Simple linear statistical methods, such as principal component analysis (PCA), fail to predict nonlinear dynamics because they represent linear combinations of mean states. Flinchem and Jay (2000) demonstrated that wavelet transforms provide an approach that is well suited for tidal phenomena that “deviate markedly from an assumed statistical stationarity or exact periodicity inherent in traditional tidal methods” and that it can also reveal features that harmonic analysis could not elucidate. In the same context, neural networks (NNs) have been used for (extreme) tidal surge prediction (French et al., 2017; Tayel and Oumeraci, 2015; and Bruneau et al., 2020). Similarly, Hieronymus et al. (2019) applied different machine learning techniques to the regression problem of time series data from tide gauges, and Balogun and Adebisi (2021) investigated the impact of different ocean–atmosphere interactions on sea-level predictability using different NNs. However, all these applications focus on data from single tide gauge stations.

Horsburgh and Wilson (2007) proved the importance of spatial and temporal correlations among tide gauge stations along the British coast for the distribution of surge residuals. However, their linear model is only capable of describing some of the possible interactions between tides and surges and was used to demonstrate the existence of critical spatiotemporal scales for surge development and decay. Wenzel and Schröter (2010) used neural networks for the reconstruction of monthly regional mean sea-level anomalies from 59 tide gauges worldwide. Zhang et al. (2020) used generative adversarial networks to reconstruct the sea-level variability in the North Sea using observations from 19 coastal tide gauges and data from numerical models. Notably, they concluded that a relatively short-term record contains the most representative characteristics of sea-level dynamics in the North Sea. While this was clear for tides, it was not so obvious for changes in sea level caused by the atmosphere. Similarly, the deep learning approach to fuse altimeter data with tide gauge data in the Mediterranean Sea was described by Yang et al. (2021), while Nieves et al. (2021) used open ocean temperatures to predict coastal sea-level variability via machine learning.

In contrast to these applications, we will focus on the identification of situations in which spatial correlations between tide gauge measurements deviate greatly from the principal correlations. Usually, tide gauge measurements in the North Sea show specific spatial correlations (changing in time with the tide). However, in anomalous situations (e.g., localized storms), these correlations may drastically change. We use autoassociative neural networks (AANNs, Kramer, 1992) to detect such sea-level states. Similar to PCA, the aim of this method is dimensionality reduction; however, the concept of orthogonal vectors is expanded to principal curves. The combination of nonlinear components best describing the variability in the training data can be used to reconstruct the data. Thus, AANN provides a nonlinear reconstruction model, and the reconstruction error is a measure of how well the data are characterized. Large errors in the reconstruction of data by the AANN represents situations in which the data do not belong to the same distribution they were built from. Atmospheric conditions related to such situations might aid further understanding and future developments in sea-level prediction.

The paper is structured as follows: in Sect. 2, we present observational and model data used throughout our study and introduce the concept of AANN. We then apply the AANN to the tide gauge array measurements in the North Sea. In Sect. 3, we use AANN to detect anomalous events and examine the dependence of identification of such events from the AANN training data used. Two events are studied in detail, including their relationship with the atmospheric conditions. This is followed by a discussion and conclusions.

To isolate events that strongly affect the spatial correlations among tide gauges, further constraints are needed. High values of AANN_ref reconstruction errors might originate from events on small scales, such as ocean response to wind gusts, as well as the erroneous measurement of a single tide gauge. From the analysis of the training period, we postulate an ocean state as anomalous when the error e exceeds 0.035 (see Fig. 4c, d) for at least three gauges simultaneously for at least 3 h.

To exclude the dependence of identification of such extreme events from the data used, we trained alternative AANNs based on the following datasets:

trained with data extracted at the closest positions of gauge stations from the CMEMS operational model AMM15 (AANN_NEMO);

The first additional network AANN_NEMO allows an intercomparison between observations and simulations. If the agreement is good, spatial data from the operational model will enable the analysis of horizontal patterns associated with the identified events. The second network, AANN_less, identifies the possible occurrence of anomalous events in the southwestern North Sea area. This area is known to be dominated by a small amphidromic system (Fig. 1a) and is characterized by its complex dynamics and high amplitudes of tidal oscillations. The third network, AANN_resid, reveals how strongly anomalous events are associated with the basic tidal dynamics, which are removed here.

The error distributions of AANN_less, AANN_resid, and AANN_NEMO are similar to those of AANN_ref. The tail of the AANN_NEMO error distribution is shortened, making modeling data less suitable for novelty detection (see Fig. 5c, d). A comparison of AANN_NEMO and AANN_ref errors is shown in Fig. 6. Anomalous situations detected with AANN_ref (4, 11–14 January 2017) reflect themselves in large (exceeding 25 cm) AANN_NEMO errors but not necessarily vice versa.

The proposed criterion was tested on the reference model AANN_ref for the 2018 testing period. Several anomalous sea states were detected (Table 1). Events in the winter months occur more than twice as often as in other seasons. Afterwards, it was verified whether a specific event could be observed with AANNs trained on alternative datasets (see Table 1). Due to the short-tailed error distribution, all events can be detected with AANN_NEMO (but as a stand-alone method, it gives a larger number of false alarms). Events from autumn and winter are also detected with the alternative AANNs (AANN_resid and AANN_less). This does not apply to spring and summer, where some events are not detected with AANN_less and/or by AANN_resid.

In the following, we will analyze two such events: the first event from June 2017 (low pressure “Heinrich”) was chosen since it was detected by AANN_resid but not by AANN_less (Figs. 7 and, 8c, d). This atmospheric system brought some unseasonably wet and windy weather from France to the British southeast coast through the Channel. The rain was accompanied by very strong westerly winds (∼15 m s-1), with gusts of 20–25 m s-1 around the coast of England and Wales (see Figs. 7a, b and 8c, d). As a result, the measured sea level (Fig. 8a) exceeded the AANN_ref modeled sea level over several hours along the western Dutch coast and southeastern British coast (Fig. 9a). The AANN_ref and AANN_NEMO (Fig. 9b) errors are largest at Oostende, where the measured low tide is higher than expected by the AANNs. The occurrence of the largest error shifts to Dover for AANN_resid and to Den Helder for PCA (Fig. 9e, f). The NEMO model error (Fig. 9c) is the largest for Vlissingen, where it under- and overestimates the measured values for high and low tides, respectively. Consistently, AANN_less does not detect the event (Fig. 9d), indicating its origin in anomalous spatial correlations among gauges around the small amphidromic system. Thus, the specific forcing resulted in a localized (exceptional) increase in sea level at gauge station Den Helder. The snapshot in Fig. 7c visualizes this finding at the time the AANN largest errors occurred, 10 h after the wind tendency.

As a second example, a typical winter storm was chosen: the deep depression “Burglind” (Figs. 10a, b and 11c, d), which was associated with high wind speeds (up to 20 m s-1) along the English and the Dutch coasts but also with large gradients in the wind field (Fig. 7a) with gusts exceeding 33 m s-1 in England and Wales. The storm track passed over Ireland, crossed the United Kingdom, and then moved over central Europe and dissipated.

As a result, the measured tide gauge values exceeded the AANN_ref, AANN_NEMO, and AANN_less values at stations along the southern English and Dutch coasts (Fig. 12a, b, d). Exceptions are Lowestoft and Cromer in the “shadow” of the cyclone. The NEMO model results (Fig. 12c) show good agreement during high tide, especially along the Scottish coast, but tend to overestimate the low tide values, especially along the Dutch coast.

AANN_resid and PCA (Fig. 12e, f) also detect the event. However, here, stations along the Dutch coast have the largest reconstruction errors (i.e., stations with the highest water level residuals, see Fig. 11b). The snapshot in Fig. 10c visualizes water level residuals at the time of the largest AANN errors, which occurred with a 7 h delay to wind tendency.

On either occasion, a typical time delay of 5 to 15 h between large tendencies in wind forcing and large reconstruction errors is observed, supporting the assumption that the atmosphere plays a key role in the detected events. However, their occurrence cannot be directly linked to atmospheric variables, such as large tendencies in the wind forcing. These tendencies occur often but seldom result in such anomalous correlations between tide gauge measurements.

We presented a tool for detecting anomalous water levels from a network of tide gauges in the North Sea using AANN. Strong spatial correlations between gauges allow for a reconstruction model in a lower dimensional subspace. Combining a threshold value for reconstruction error with the requirement of its occurrence at three different gauges for at least three hours has been shown to be a valuable filter for such events. For the two events discussed in detail, atmospheric conditions showed high wind tendencies and pressure anomalies, with the accumulation of water masses in specific areas. Thus, the resultant sea states are assumed to be related to nonlinear interactions among the various atmospheric and oceanic forcings. Further analysis is needed to develop a deeper understanding of the underlying processes. Ultimately, this understanding might improve numerical sea-level predictions. The AANN reveals an intrinsic dimensionality of p=7. Evident variable candidates that could be involved in further studies include location (latitude/longitude), time, wind tendency, pressure anomaly, and information on water currents.

In addition, the tool offers an inexpensive opportunity to monitor the tidal gauge array. The difference between using raw and detided data as input to the model was marginal. Thus, no further data preprocessing is needed. The AANN model reacts sensitively to the choice of tide gauge locations: AANN_less, where stations around the small amphidromic system were removed from the training data, was not able to identify events resulting from storms along the channel.

The linear PCA reconstruction method reveals a high rate of false positive alarms, which indicates a multimodal probability distribution of the sea-level data. The detection of some of these events might fail for AANN_NEMO. This finding substantiates the hypothesis that AANN is able to detect situations under which model physics need further improvement.

The proposed method can be adapted easily to any tide gauge array. However, the intrinsic dimensionality of the constructed AANN might differ, as well as the involved forcings and underlying processes.